Kervaire invariant problem

The Kervaire invariant is an invariant of a framed (4k+2)-dimensional manifold that measures whether the manifold could be surgically converted into a sphere. This invariant evaluates to 0 if the manifold can be converted to a sphere, and 1 otherwise. This invariant was named after Michel Kervaire who built on work of Cahit Arf.

The Kervaire invariant is defined as the Arf invariant of the skew-quadratic form on the middle dimensional homology group. It can be thought of as the simply-connected quadratic L-group ${\displaystyle L_{4k+2},}$ and thus analogous to the other invariants from L-theory: the signature, a 4k-dimensional invariant (either symmetric or quadratic, ${\displaystyle L^{4k}\cong L_{4k}}$), and the De Rham invariant, a (4k+1)-dimensional symmetric invariant ${\displaystyle L^{4k+1}.}$

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